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x/(x-1)^2

Derivative of x/(x-1)^2

Function f() - derivative -N order at the point
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The graph:

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The solution

You have entered [src]
   x    
--------
       2
(x - 1) 
x(x1)2\frac{x}{\left(x - 1\right)^{2}}
x/(x - 1)^2
Detail solution
  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

    f(x)=xf{\left(x \right)} = x and g(x)=(x1)2g{\left(x \right)} = \left(x - 1\right)^{2}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: xx goes to 11

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=x1u = x - 1.

    2. Apply the power rule: u2u^{2} goes to 2u2 u

    3. Then, apply the chain rule. Multiply by ddx(x1)\frac{d}{d x} \left(x - 1\right):

      1. Differentiate x1x - 1 term by term:

        1. The derivative of the constant 1-1 is zero.

        2. Apply the power rule: xx goes to 11

        The result is: 11

      The result of the chain rule is:

      2x22 x - 2

    Now plug in to the quotient rule:

    x(2x2)+(x1)2(x1)4\frac{- x \left(2 x - 2\right) + \left(x - 1\right)^{2}}{\left(x - 1\right)^{4}}

  2. Now simplify:

    x+1(x1)3- \frac{x + 1}{\left(x - 1\right)^{3}}


The answer is:

x+1(x1)3- \frac{x + 1}{\left(x - 1\right)^{3}}

The graph
02468-8-6-4-2-1010-50005000
The first derivative [src]
   1       x*(2 - 2*x)
-------- + -----------
       2            4 
(x - 1)      (x - 1)  
x(22x)(x1)4+1(x1)2\frac{x \left(2 - 2 x\right)}{\left(x - 1\right)^{4}} + \frac{1}{\left(x - 1\right)^{2}}
The second derivative [src]
  /      3*x  \
2*|-2 + ------|
  \     -1 + x/
---------------
           3   
   (-1 + x)    
2(3xx12)(x1)3\frac{2 \left(\frac{3 x}{x - 1} - 2\right)}{\left(x - 1\right)^{3}}
The third derivative [src]
  /     4*x  \
6*|3 - ------|
  \    -1 + x/
--------------
          4   
  (-1 + x)    
6(4xx1+3)(x1)4\frac{6 \left(- \frac{4 x}{x - 1} + 3\right)}{\left(x - 1\right)^{4}}
The graph
Derivative of x/(x-1)^2